Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B

Percentage Accurate: 96.6% → 96.0%

Time: 12.6s

Alternatives: 11

Speedup: N/A×

• Could not converge on a ground truth

  1. x = 4.520496430286436e-296
  2. y = 2.8802368426348095e-303
  3. z = 3.9431583075889366e-204
  4. t = 4.234151990415885e+173
  5. a = -3.6883056491937807e-62
  6. b = -1.5797837343945954e-205

• 42.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))));
}

Python

def code(x, y, z, t, a, b):
	return x * math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 96.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))));
}

Python

def code(x, y, z, t, a, b):
	return x * math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.5 \cdot 10^{+166}:\\ \;\;\;\;e^{\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{-\mathsf{fma}\left(a, z, b \cdot a\right)} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a 2.5e+166)
   (* (exp (+ (* (- (log (- 1.0 z)) b) a) (* (- (log z) t) y))) x)
   (* (exp (- (fma a z (* b a)))) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= 2.5e+166) {
		tmp = exp((((log((1.0 - z)) - b) * a) + ((log(z) - t) * y))) * x;
	} else {
		tmp = exp(-fma(a, z, (b * a))) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= 2.5e+166)
		tmp = Float64(exp(Float64(Float64(Float64(log(Float64(1.0 - z)) - b) * a) + Float64(Float64(log(z) - t) * y))) * x);
	else
		tmp = Float64(exp(Float64(-fma(a, z, Float64(b * a)))) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, 2.5e+166], N[(N[Exp[N[(N[(N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], N[(N[Exp[(-N[(a * z + N[(b * a), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.5000000000000001e166

   1. Initial program 95.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   if 2.5000000000000001e166 < a

   1. Initial program 74.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      3. lower--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right)} \cdot a} \]

      4. sub-neg N/A

         \[\leadsto x \cdot e^{\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right) \cdot a} \]

      5. lower-log1p.f64 N/A

         \[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right) \cdot a} \]

      6. lower-neg.f64 91.6

         \[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a} \]

   5. Applied rewrites 91.6%

      \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{-1 \cdot \left(a \cdot b\right) + \color{blue}{-1 \cdot \left(a \cdot z\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 91.6%

      \[\leadsto x \cdot e^{-\mathsf{fma}\left(a, z, a \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.5 \cdot 10^{+166}:\\ \;\;\;\;e^{\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{-\mathsf{fma}\left(a, z, b \cdot a\right)} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 49.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(0.5 \cdot t\right) \cdot \left(\left(y \cdot y\right) \cdot x\right)\right) \cdot t\\ t_2 := \left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* 0.5 t) (* (* y y) x)) t))
        (t_2 (+ (* (- (log (- 1.0 z)) b) a) (* (- (log z) t) y))))
   (if (<= t_2 -2e+29) t_1 (if (<= t_2 2e-13) (* 1.0 x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((0.5 * t) * ((y * y) * x)) * t;
	double t_2 = ((log((1.0 - z)) - b) * a) + ((log(z) - t) * y);
	double tmp;
	if (t_2 <= -2e+29) {
		tmp = t_1;
	} else if (t_2 <= 2e-13) {
		tmp = 1.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((0.5d0 * t) * ((y * y) * x)) * t
    t_2 = ((log((1.0d0 - z)) - b) * a) + ((log(z) - t) * y)
    if (t_2 <= (-2d+29)) then
        tmp = t_1
    else if (t_2 <= 2d-13) then
        tmp = 1.0d0 * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((0.5 * t) * ((y * y) * x)) * t;
	double t_2 = ((Math.log((1.0 - z)) - b) * a) + ((Math.log(z) - t) * y);
	double tmp;
	if (t_2 <= -2e+29) {
		tmp = t_1;
	} else if (t_2 <= 2e-13) {
		tmp = 1.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = ((0.5 * t) * ((y * y) * x)) * t
	t_2 = ((math.log((1.0 - z)) - b) * a) + ((math.log(z) - t) * y)
	tmp = 0
	if t_2 <= -2e+29:
		tmp = t_1
	elif t_2 <= 2e-13:
		tmp = 1.0 * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(0.5 * t) * Float64(Float64(y * y) * x)) * t)
	t_2 = Float64(Float64(Float64(log(Float64(1.0 - z)) - b) * a) + Float64(Float64(log(z) - t) * y))
	tmp = 0.0
	if (t_2 <= -2e+29)
		tmp = t_1;
	elseif (t_2 <= 2e-13)
		tmp = Float64(1.0 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((0.5 * t) * ((y * y) * x)) * t;
	t_2 = ((log((1.0 - z)) - b) * a) + ((log(z) - t) * y);
	tmp = 0.0;
	if (t_2 <= -2e+29)
		tmp = t_1;
	elseif (t_2 <= 2e-13)
		tmp = 1.0 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(0.5 * t), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+29], t$95$1, If[LessEqual[t$95$2, 2e-13], N[(1.0 * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1.99999999999999983e29 or 2.0000000000000001e-13 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 94.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)} + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right)\right)} \]

   5. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left({\left(\frac{1 - z}{e^{b}}\right)}^{a} \cdot \mathsf{fma}\left(0.5 \cdot y, {\left(\log z - t\right)}^{2}, \log z - t\right)\right), y, {\left(\frac{1 - z}{e^{b}}\right)}^{a} \cdot x\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{x \cdot \left(y \cdot \left(\left(\log z + \frac{1}{2} \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) - t\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.8%

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left({\left(\log z - t\right)}^{2} \cdot y, 0.5, \log z\right) - t\right) \cdot y, \color{blue}{x}, x\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \frac{1}{2} \cdot \left({t}^{2} \cdot \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 34.6%

       \[\leadsto \left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{x}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 40.1%

       \[\leadsto \left(\left(\left(y \cdot y\right) \cdot x\right) \cdot \left(0.5 \cdot t\right)\right) \cdot t \]

   if -1.99999999999999983e29 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2.0000000000000001e-13

   1. Initial program 84.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{e^{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}} \]

      2. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      3. lower-pow.f64 N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      4. exp-diff N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{e^{\log z}}{e^{t}}\right)}}^{y} \]

      5. rem-exp-log N/A

         \[\leadsto x \cdot {\left(\frac{\color{blue}{z}}{e^{t}}\right)}^{y} \]

      6. lower-/.f64 N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{z}{e^{t}}\right)}}^{y} \]

      7. lower-exp.f64 59.5

         \[\leadsto x \cdot {\left(\frac{z}{\color{blue}{e^{t}}}\right)}^{y} \]

   5. Applied rewrites 59.5%

      \[\leadsto x \cdot \color{blue}{{\left(\frac{z}{e^{t}}\right)}^{y}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 80.1%

      \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq -2 \cdot 10^{+29}:\\ \;\;\;\;\left(\left(0.5 \cdot t\right) \cdot \left(\left(y \cdot y\right) \cdot x\right)\right) \cdot t\\ \mathbf{elif}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq 2 \cdot 10^{-13}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.5 \cdot t\right) \cdot \left(\left(y \cdot y\right) \cdot x\right)\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y\\ t_2 := \left(t \cdot t\right) \cdot 0.5\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+39}:\\ \;\;\;\;\left(t\_2 \cdot \left(y \cdot x\right)\right) \cdot y\\ \mathbf{elif}\;t\_1 \leq 0.04:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 \cdot x\right) \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* (- (log (- 1.0 z)) b) a) (* (- (log z) t) y)))
        (t_2 (* (* t t) 0.5)))
   (if (<= t_1 -1e+39)
     (* (* t_2 (* y x)) y)
     (if (<= t_1 0.04) (* 1.0 x) (* (* t_2 x) (* y y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((log((1.0 - z)) - b) * a) + ((log(z) - t) * y);
	double t_2 = (t * t) * 0.5;
	double tmp;
	if (t_1 <= -1e+39) {
		tmp = (t_2 * (y * x)) * y;
	} else if (t_1 <= 0.04) {
		tmp = 1.0 * x;
	} else {
		tmp = (t_2 * x) * (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((log((1.0d0 - z)) - b) * a) + ((log(z) - t) * y)
    t_2 = (t * t) * 0.5d0
    if (t_1 <= (-1d+39)) then
        tmp = (t_2 * (y * x)) * y
    else if (t_1 <= 0.04d0) then
        tmp = 1.0d0 * x
    else
        tmp = (t_2 * x) * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((Math.log((1.0 - z)) - b) * a) + ((Math.log(z) - t) * y);
	double t_2 = (t * t) * 0.5;
	double tmp;
	if (t_1 <= -1e+39) {
		tmp = (t_2 * (y * x)) * y;
	} else if (t_1 <= 0.04) {
		tmp = 1.0 * x;
	} else {
		tmp = (t_2 * x) * (y * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = ((math.log((1.0 - z)) - b) * a) + ((math.log(z) - t) * y)
	t_2 = (t * t) * 0.5
	tmp = 0
	if t_1 <= -1e+39:
		tmp = (t_2 * (y * x)) * y
	elif t_1 <= 0.04:
		tmp = 1.0 * x
	else:
		tmp = (t_2 * x) * (y * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(log(Float64(1.0 - z)) - b) * a) + Float64(Float64(log(z) - t) * y))
	t_2 = Float64(Float64(t * t) * 0.5)
	tmp = 0.0
	if (t_1 <= -1e+39)
		tmp = Float64(Float64(t_2 * Float64(y * x)) * y);
	elseif (t_1 <= 0.04)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(t_2 * x) * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((log((1.0 - z)) - b) * a) + ((log(z) - t) * y);
	t_2 = (t * t) * 0.5;
	tmp = 0.0;
	if (t_1 <= -1e+39)
		tmp = (t_2 * (y * x)) * y;
	elseif (t_1 <= 0.04)
		tmp = 1.0 * x;
	else
		tmp = (t_2 * x) * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * t), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+39], N[(N[(t$95$2 * N[(y * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 0.04], N[(1.0 * x), $MachinePrecision], N[(N[(t$95$2 * x), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -9.9999999999999994e38

   1. Initial program 99.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)} + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right)\right)} \]

   5. Applied rewrites 35.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left({\left(\frac{1 - z}{e^{b}}\right)}^{a} \cdot \mathsf{fma}\left(0.5 \cdot y, {\left(\log z - t\right)}^{2}, \log z - t\right)\right), y, {\left(\frac{1 - z}{e^{b}}\right)}^{a} \cdot x\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{x \cdot \left(y \cdot \left(\left(\log z + \frac{1}{2} \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) - t\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.2%

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left({\left(\log z - t\right)}^{2} \cdot y, 0.5, \log z\right) - t\right) \cdot y, \color{blue}{x}, x\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \frac{1}{2} \cdot \left({t}^{2} \cdot \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 29.8%

       \[\leadsto \left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{x}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 30.3%

       \[\leadsto y \cdot \left(\left(y \cdot x\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{0.5}\right)\right) \]

   if -9.9999999999999994e38 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 0.0400000000000000008

   1. Initial program 83.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{e^{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}} \]

      2. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      3. lower-pow.f64 N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      4. exp-diff N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{e^{\log z}}{e^{t}}\right)}}^{y} \]

      5. rem-exp-log N/A

         \[\leadsto x \cdot {\left(\frac{\color{blue}{z}}{e^{t}}\right)}^{y} \]

      6. lower-/.f64 N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{z}{e^{t}}\right)}}^{y} \]

      7. lower-exp.f64 57.2

         \[\leadsto x \cdot {\left(\frac{z}{\color{blue}{e^{t}}}\right)}^{y} \]

   5. Applied rewrites 57.2%

      \[\leadsto x \cdot \color{blue}{{\left(\frac{z}{e^{t}}\right)}^{y}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 74.0%

      \[\leadsto x \cdot 1 \]

   if 0.0400000000000000008 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 90.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)} + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right)\right)} \]

   5. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left({\left(\frac{1 - z}{e^{b}}\right)}^{a} \cdot \mathsf{fma}\left(0.5 \cdot y, {\left(\log z - t\right)}^{2}, \log z - t\right)\right), y, {\left(\frac{1 - z}{e^{b}}\right)}^{a} \cdot x\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{x \cdot \left(y \cdot \left(\left(\log z + \frac{1}{2} \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) - t\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.5%

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left({\left(\log z - t\right)}^{2} \cdot y, 0.5, \log z\right) - t\right) \cdot y, \color{blue}{x}, x\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \frac{1}{2} \cdot \left({t}^{2} \cdot \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 40.2%

       \[\leadsto \left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{x}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 43.8%

       \[\leadsto \left(\left(\left(t \cdot t\right) \cdot 0.5\right) \cdot x\right) \cdot \left(y \cdot y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq -1 \cdot 10^{+39}:\\ \;\;\;\;\left(\left(\left(t \cdot t\right) \cdot 0.5\right) \cdot \left(y \cdot x\right)\right) \cdot y\\ \mathbf{elif}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq 0.04:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot t\right) \cdot 0.5\right) \cdot x\right) \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 45.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(t \cdot t\right) \cdot 0.5\right) \cdot \left(y \cdot x\right)\right) \cdot y\\ t_2 := \left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.04:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* (* t t) 0.5) (* y x)) y))
        (t_2 (+ (* (- (log (- 1.0 z)) b) a) (* (- (log z) t) y))))
   (if (<= t_2 -1e+39) t_1 (if (<= t_2 0.04) (* 1.0 x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((t * t) * 0.5) * (y * x)) * y;
	double t_2 = ((log((1.0 - z)) - b) * a) + ((log(z) - t) * y);
	double tmp;
	if (t_2 <= -1e+39) {
		tmp = t_1;
	} else if (t_2 <= 0.04) {
		tmp = 1.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (((t * t) * 0.5d0) * (y * x)) * y
    t_2 = ((log((1.0d0 - z)) - b) * a) + ((log(z) - t) * y)
    if (t_2 <= (-1d+39)) then
        tmp = t_1
    else if (t_2 <= 0.04d0) then
        tmp = 1.0d0 * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((t * t) * 0.5) * (y * x)) * y;
	double t_2 = ((Math.log((1.0 - z)) - b) * a) + ((Math.log(z) - t) * y);
	double tmp;
	if (t_2 <= -1e+39) {
		tmp = t_1;
	} else if (t_2 <= 0.04) {
		tmp = 1.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (((t * t) * 0.5) * (y * x)) * y
	t_2 = ((math.log((1.0 - z)) - b) * a) + ((math.log(z) - t) * y)
	tmp = 0
	if t_2 <= -1e+39:
		tmp = t_1
	elif t_2 <= 0.04:
		tmp = 1.0 * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(t * t) * 0.5) * Float64(y * x)) * y)
	t_2 = Float64(Float64(Float64(log(Float64(1.0 - z)) - b) * a) + Float64(Float64(log(z) - t) * y))
	tmp = 0.0
	if (t_2 <= -1e+39)
		tmp = t_1;
	elseif (t_2 <= 0.04)
		tmp = Float64(1.0 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (((t * t) * 0.5) * (y * x)) * y;
	t_2 = ((log((1.0 - z)) - b) * a) + ((log(z) - t) * y);
	tmp = 0.0;
	if (t_2 <= -1e+39)
		tmp = t_1;
	elseif (t_2 <= 0.04)
		tmp = 1.0 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(t * t), $MachinePrecision] * 0.5), $MachinePrecision] * N[(y * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+39], t$95$1, If[LessEqual[t$95$2, 0.04], N[(1.0 * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -9.9999999999999994e38 or 0.0400000000000000008 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 94.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)} + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right)\right)} \]

   5. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left({\left(\frac{1 - z}{e^{b}}\right)}^{a} \cdot \mathsf{fma}\left(0.5 \cdot y, {\left(\log z - t\right)}^{2}, \log z - t\right)\right), y, {\left(\frac{1 - z}{e^{b}}\right)}^{a} \cdot x\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{x \cdot \left(y \cdot \left(\left(\log z + \frac{1}{2} \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) - t\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.3%

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left({\left(\log z - t\right)}^{2} \cdot y, 0.5, \log z\right) - t\right) \cdot y, \color{blue}{x}, x\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \frac{1}{2} \cdot \left({t}^{2} \cdot \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 35.2%

       \[\leadsto \left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{x}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 36.5%

       \[\leadsto y \cdot \left(\left(y \cdot x\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{0.5}\right)\right) \]

   if -9.9999999999999994e38 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 0.0400000000000000008

   1. Initial program 83.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{e^{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}} \]

      2. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      3. lower-pow.f64 N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      4. exp-diff N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{e^{\log z}}{e^{t}}\right)}}^{y} \]

      5. rem-exp-log N/A

         \[\leadsto x \cdot {\left(\frac{\color{blue}{z}}{e^{t}}\right)}^{y} \]

      6. lower-/.f64 N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{z}{e^{t}}\right)}}^{y} \]

      7. lower-exp.f64 57.2

         \[\leadsto x \cdot {\left(\frac{z}{\color{blue}{e^{t}}}\right)}^{y} \]

   5. Applied rewrites 57.2%

      \[\leadsto x \cdot \color{blue}{{\left(\frac{z}{e^{t}}\right)}^{y}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 74.0%

      \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq -1 \cdot 10^{+39}:\\ \;\;\;\;\left(\left(\left(t \cdot t\right) \cdot 0.5\right) \cdot \left(y \cdot x\right)\right) \cdot y\\ \mathbf{elif}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq 0.04:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot t\right) \cdot 0.5\right) \cdot \left(y \cdot x\right)\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{z}{e^{t}}\right)}^{y} \cdot x\\ \mathbf{if}\;y \leq -0.0001:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+27}:\\ \;\;\;\;e^{-\mathsf{fma}\left(a, z, b \cdot a\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (pow (/ z (exp t)) y) x)))
   (if (<= y -0.0001)
     t_1
     (if (<= y 3.2e+27) (* (exp (- (fma a z (* b a)))) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = pow((z / exp(t)), y) * x;
	double tmp;
	if (y <= -0.0001) {
		tmp = t_1;
	} else if (y <= 3.2e+27) {
		tmp = exp(-fma(a, z, (b * a))) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64((Float64(z / exp(t)) ^ y) * x)
	tmp = 0.0
	if (y <= -0.0001)
		tmp = t_1;
	elseif (y <= 3.2e+27)
		tmp = Float64(exp(Float64(-fma(a, z, Float64(b * a)))) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Power[N[(z / N[Exp[t], $MachinePrecision]), $MachinePrecision], y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -0.0001], t$95$1, If[LessEqual[y, 3.2e+27], N[(N[Exp[(-N[(a * z + N[(b * a), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000005e-4 or 3.20000000000000015e27 < y

   1. Initial program 94.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{e^{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}} \]

      2. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      3. lower-pow.f64 N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      4. exp-diff N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{e^{\log z}}{e^{t}}\right)}}^{y} \]

      5. rem-exp-log N/A

         \[\leadsto x \cdot {\left(\frac{\color{blue}{z}}{e^{t}}\right)}^{y} \]

      6. lower-/.f64 N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{z}{e^{t}}\right)}}^{y} \]

      7. lower-exp.f64 84.1

         \[\leadsto x \cdot {\left(\frac{z}{\color{blue}{e^{t}}}\right)}^{y} \]

   5. Applied rewrites 84.1%

      \[\leadsto x \cdot \color{blue}{{\left(\frac{z}{e^{t}}\right)}^{y}} \]

   if -1.00000000000000005e-4 < y < 3.20000000000000015e27

   1. Initial program 91.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      3. lower--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right)} \cdot a} \]

      4. sub-neg N/A

         \[\leadsto x \cdot e^{\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right) \cdot a} \]

      5. lower-log1p.f64 N/A

         \[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right) \cdot a} \]

      6. lower-neg.f64 89.1

         \[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a} \]

   5. Applied rewrites 89.1%

      \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{-1 \cdot \left(a \cdot b\right) + \color{blue}{-1 \cdot \left(a \cdot z\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 89.1%

      \[\leadsto x \cdot e^{-\mathsf{fma}\left(a, z, a \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0001:\\ \;\;\;\;{\left(\frac{z}{e^{t}}\right)}^{y} \cdot x\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+27}:\\ \;\;\;\;e^{-\mathsf{fma}\left(a, z, b \cdot a\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{z}{e^{t}}\right)}^{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-\mathsf{fma}\left(a, z, b \cdot a\right)} \cdot x\\ \mathbf{if}\;a \leq -3.4 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-101}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (- (fma a z (* b a)))) x)))
   (if (<= a -3.4e-14) t_1 (if (<= a 5.5e-101) (* (exp (* (- t) y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp(-fma(a, z, (b * a))) * x;
	double tmp;
	if (a <= -3.4e-14) {
		tmp = t_1;
	} else if (a <= 5.5e-101) {
		tmp = exp((-t * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(-fma(a, z, Float64(b * a)))) * x)
	tmp = 0.0
	if (a <= -3.4e-14)
		tmp = t_1;
	elseif (a <= 5.5e-101)
		tmp = Float64(exp(Float64(Float64(-t) * y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[(-N[(a * z + N[(b * a), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[a, -3.4e-14], t$95$1, If[LessEqual[a, 5.5e-101], N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.40000000000000003e-14 or 5.49999999999999973e-101 < a

   1. Initial program 89.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      3. lower--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right)} \cdot a} \]

      4. sub-neg N/A

         \[\leadsto x \cdot e^{\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right) \cdot a} \]

      5. lower-log1p.f64 N/A

         \[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right) \cdot a} \]

      6. lower-neg.f64 82.3

         \[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a} \]

   5. Applied rewrites 82.3%

      \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{-1 \cdot \left(a \cdot b\right) + \color{blue}{-1 \cdot \left(a \cdot z\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 82.3%

      \[\leadsto x \cdot e^{-\mathsf{fma}\left(a, z, a \cdot b\right)} \]

   if -3.40000000000000003e-14 < a < 5.49999999999999973e-101

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot t\right) \cdot y}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{neg}\left(t\right)\right)} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{neg}\left(t\right)\right) \cdot y}} \]

      4. lower-neg.f64 73.5

         \[\leadsto x \cdot e^{\color{blue}{\left(-t\right)} \cdot y} \]

   5. Applied rewrites 73.5%

      \[\leadsto x \cdot e^{\color{blue}{\left(-t\right) \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{-14}:\\ \;\;\;\;e^{-\mathsf{fma}\left(a, z, b \cdot a\right)} \cdot x\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-101}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{-\mathsf{fma}\left(a, z, b \cdot a\right)} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{if}\;b \leq -1.2 \cdot 10^{-69}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 7800000000000:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- b) a)) x)))
   (if (<= b -1.2e-69)
     t_1
     (if (<= b 7800000000000.0) (* (exp (* (- t) y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((-b * a)) * x;
	double tmp;
	if (b <= -1.2e-69) {
		tmp = t_1;
	} else if (b <= 7800000000000.0) {
		tmp = exp((-t * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp((-b * a)) * x
    if (b <= (-1.2d-69)) then
        tmp = t_1
    else if (b <= 7800000000000.0d0) then
        tmp = exp((-t * y)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp((-b * a)) * x;
	double tmp;
	if (b <= -1.2e-69) {
		tmp = t_1;
	} else if (b <= 7800000000000.0) {
		tmp = Math.exp((-t * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((-b * a)) * x
	tmp = 0
	if b <= -1.2e-69:
		tmp = t_1
	elif b <= 7800000000000.0:
		tmp = math.exp((-t * y)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(-b) * a)) * x)
	tmp = 0.0
	if (b <= -1.2e-69)
		tmp = t_1;
	elseif (b <= 7800000000000.0)
		tmp = Float64(exp(Float64(Float64(-t) * y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((-b * a)) * x;
	tmp = 0.0;
	if (b <= -1.2e-69)
		tmp = t_1;
	elseif (b <= 7800000000000.0)
		tmp = exp((-t * y)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[b, -1.2e-69], t$95$1, If[LessEqual[b, 7800000000000.0], N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.2000000000000001e-69 or 7.8e12 < b

   1. Initial program 96.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      3. lower--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right)} \cdot a} \]

      4. sub-neg N/A

         \[\leadsto x \cdot e^{\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right) \cdot a} \]

      5. lower-log1p.f64 N/A

         \[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right) \cdot a} \]

      6. lower-neg.f64 80.4

         \[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a} \]

   5. Applied rewrites 80.4%

      \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{-1 \cdot \color{blue}{\left(a \cdot b\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.3%

      \[\leadsto x \cdot e^{\left(-b\right) \cdot \color{blue}{a}} \]

   if -1.2000000000000001e-69 < b < 7.8e12

   1. Initial program 88.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot t\right) \cdot y}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{neg}\left(t\right)\right)} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{neg}\left(t\right)\right) \cdot y}} \]

      4. lower-neg.f64 65.2

         \[\leadsto x \cdot e^{\color{blue}{\left(-t\right)} \cdot y} \]

   5. Applied rewrites 65.2%

      \[\leadsto x \cdot e^{\color{blue}{\left(-t\right) \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-69}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{elif}\;b \leq 7800000000000:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{if}\;b \leq -2.4 \cdot 10^{-69}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{+53}:\\ \;\;\;\;{z}^{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- b) a)) x)))
   (if (<= b -2.4e-69) t_1 (if (<= b 2.8e+53) (* (pow z y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((-b * a)) * x;
	double tmp;
	if (b <= -2.4e-69) {
		tmp = t_1;
	} else if (b <= 2.8e+53) {
		tmp = pow(z, y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp((-b * a)) * x
    if (b <= (-2.4d-69)) then
        tmp = t_1
    else if (b <= 2.8d+53) then
        tmp = (z ** y) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp((-b * a)) * x;
	double tmp;
	if (b <= -2.4e-69) {
		tmp = t_1;
	} else if (b <= 2.8e+53) {
		tmp = Math.pow(z, y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((-b * a)) * x
	tmp = 0
	if b <= -2.4e-69:
		tmp = t_1
	elif b <= 2.8e+53:
		tmp = math.pow(z, y) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(-b) * a)) * x)
	tmp = 0.0
	if (b <= -2.4e-69)
		tmp = t_1;
	elseif (b <= 2.8e+53)
		tmp = Float64((z ^ y) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((-b * a)) * x;
	tmp = 0.0;
	if (b <= -2.4e-69)
		tmp = t_1;
	elseif (b <= 2.8e+53)
		tmp = (z ^ y) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[b, -2.4e-69], t$95$1, If[LessEqual[b, 2.8e+53], N[(N[Power[z, y], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.4000000000000001e-69 or 2.8e53 < b

   1. Initial program 96.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      3. lower--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right)} \cdot a} \]

      4. sub-neg N/A

         \[\leadsto x \cdot e^{\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right) \cdot a} \]

      5. lower-log1p.f64 N/A

         \[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right) \cdot a} \]

      6. lower-neg.f64 82.9

         \[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a} \]

   5. Applied rewrites 82.9%

      \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{-1 \cdot \color{blue}{\left(a \cdot b\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 82.8%

      \[\leadsto x \cdot e^{\left(-b\right) \cdot \color{blue}{a}} \]

   if -2.4000000000000001e-69 < b < 2.8e53

   1. Initial program 89.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{e^{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}} \]

      2. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      3. lower-pow.f64 N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      4. exp-diff N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{e^{\log z}}{e^{t}}\right)}}^{y} \]

      5. rem-exp-log N/A

         \[\leadsto x \cdot {\left(\frac{\color{blue}{z}}{e^{t}}\right)}^{y} \]

      6. lower-/.f64 N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{z}{e^{t}}\right)}}^{y} \]

      7. lower-exp.f64 75.9

         \[\leadsto x \cdot {\left(\frac{z}{\color{blue}{e^{t}}}\right)}^{y} \]

   5. Applied rewrites 75.9%

      \[\leadsto x \cdot \color{blue}{{\left(\frac{z}{e^{t}}\right)}^{y}} \]

   6. Taylor expanded in t around 0

      \[\leadsto x \cdot {z}^{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.8%

      \[\leadsto x \cdot {z}^{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-69}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{+53}:\\ \;\;\;\;{z}^{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {z}^{y} \cdot x\\ \mathbf{if}\;y \leq -1.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-66}:\\ \;\;\;\;e^{-z \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (pow z y) x)))
   (if (<= y -1.2) t_1 (if (<= y 3.1e-66) (* (exp (- (* z a))) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = pow(z, y) * x;
	double tmp;
	if (y <= -1.2) {
		tmp = t_1;
	} else if (y <= 3.1e-66) {
		tmp = exp(-(z * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z ** y) * x
    if (y <= (-1.2d0)) then
        tmp = t_1
    else if (y <= 3.1d-66) then
        tmp = exp(-(z * a)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.pow(z, y) * x;
	double tmp;
	if (y <= -1.2) {
		tmp = t_1;
	} else if (y <= 3.1e-66) {
		tmp = Math.exp(-(z * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.pow(z, y) * x
	tmp = 0
	if y <= -1.2:
		tmp = t_1
	elif y <= 3.1e-66:
		tmp = math.exp(-(z * a)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64((z ^ y) * x)
	tmp = 0.0
	if (y <= -1.2)
		tmp = t_1;
	elseif (y <= 3.1e-66)
		tmp = Float64(exp(Float64(-Float64(z * a))) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z ^ y) * x;
	tmp = 0.0;
	if (y <= -1.2)
		tmp = t_1;
	elseif (y <= 3.1e-66)
		tmp = exp(-(z * a)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Power[z, y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -1.2], t$95$1, If[LessEqual[y, 3.1e-66], N[(N[Exp[(-N[(z * a), $MachinePrecision])], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.19999999999999996 or 3.0999999999999997e-66 < y

   1. Initial program 95.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{e^{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}} \]

      2. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      3. lower-pow.f64 N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      4. exp-diff N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{e^{\log z}}{e^{t}}\right)}}^{y} \]

      5. rem-exp-log N/A

         \[\leadsto x \cdot {\left(\frac{\color{blue}{z}}{e^{t}}\right)}^{y} \]

      6. lower-/.f64 N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{z}{e^{t}}\right)}}^{y} \]

      7. lower-exp.f64 80.2

         \[\leadsto x \cdot {\left(\frac{z}{\color{blue}{e^{t}}}\right)}^{y} \]

   5. Applied rewrites 80.2%

      \[\leadsto x \cdot \color{blue}{{\left(\frac{z}{e^{t}}\right)}^{y}} \]

   6. Taylor expanded in t around 0

      \[\leadsto x \cdot {z}^{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.6%

      \[\leadsto x \cdot {z}^{\color{blue}{y}} \]

   if -1.19999999999999996 < y < 3.0999999999999997e-66

   1. Initial program 88.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      3. lower--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right)} \cdot a} \]

      4. sub-neg N/A

         \[\leadsto x \cdot e^{\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right) \cdot a} \]

      5. lower-log1p.f64 N/A

         \[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right) \cdot a} \]

      6. lower-neg.f64 90.3

         \[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a} \]

   5. Applied rewrites 90.3%

      \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{-1 \cdot \left(a \cdot b\right) + \color{blue}{-1 \cdot \left(a \cdot z\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 90.3%

      \[\leadsto x \cdot e^{-\mathsf{fma}\left(a, z, a \cdot b\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto x \cdot e^{-a \cdot z} \]
   10. Step-by-step derivation

   11. Applied rewrites 57.0%

       \[\leadsto x \cdot e^{-z \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2:\\ \;\;\;\;{z}^{y} \cdot x\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-66}:\\ \;\;\;\;e^{-z \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;{z}^{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(0.5 \cdot t\right) \cdot \left(\left(y \cdot y\right) \cdot x\right)\right) \cdot t\\ \mathbf{if}\;a \leq -1.28 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{+166}:\\ \;\;\;\;{z}^{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* 0.5 t) (* (* y y) x)) t)))
   (if (<= a -1.28e+141) t_1 (if (<= a 1.7e+166) (* (pow z y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((0.5 * t) * ((y * y) * x)) * t;
	double tmp;
	if (a <= -1.28e+141) {
		tmp = t_1;
	} else if (a <= 1.7e+166) {
		tmp = pow(z, y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((0.5d0 * t) * ((y * y) * x)) * t
    if (a <= (-1.28d+141)) then
        tmp = t_1
    else if (a <= 1.7d+166) then
        tmp = (z ** y) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((0.5 * t) * ((y * y) * x)) * t;
	double tmp;
	if (a <= -1.28e+141) {
		tmp = t_1;
	} else if (a <= 1.7e+166) {
		tmp = Math.pow(z, y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = ((0.5 * t) * ((y * y) * x)) * t
	tmp = 0
	if a <= -1.28e+141:
		tmp = t_1
	elif a <= 1.7e+166:
		tmp = math.pow(z, y) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(0.5 * t) * Float64(Float64(y * y) * x)) * t)
	tmp = 0.0
	if (a <= -1.28e+141)
		tmp = t_1;
	elseif (a <= 1.7e+166)
		tmp = Float64((z ^ y) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((0.5 * t) * ((y * y) * x)) * t;
	tmp = 0.0;
	if (a <= -1.28e+141)
		tmp = t_1;
	elseif (a <= 1.7e+166)
		tmp = (z ^ y) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(0.5 * t), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[a, -1.28e+141], t$95$1, If[LessEqual[a, 1.7e+166], N[(N[Power[z, y], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.28000000000000004e141 or 1.7e166 < a

   1. Initial program 84.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)} + y \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right) + x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right)\right)} \]

   5. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left({\left(\frac{1 - z}{e^{b}}\right)}^{a} \cdot \mathsf{fma}\left(0.5 \cdot y, {\left(\log z - t\right)}^{2}, \log z - t\right)\right), y, {\left(\frac{1 - z}{e^{b}}\right)}^{a} \cdot x\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{x \cdot \left(y \cdot \left(\left(\log z + \frac{1}{2} \cdot \left(y \cdot {\left(\log z - t\right)}^{2}\right)\right) - t\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 27.1%

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left({\left(\log z - t\right)}^{2} \cdot y, 0.5, \log z\right) - t\right) \cdot y, \color{blue}{x}, x\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \frac{1}{2} \cdot \left({t}^{2} \cdot \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 42.3%

       \[\leadsto \left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{x}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 50.1%

       \[\leadsto \left(\left(\left(y \cdot y\right) \cdot x\right) \cdot \left(0.5 \cdot t\right)\right) \cdot t \]

   if -1.28000000000000004e141 < a < 1.7e166

   1. Initial program 96.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{e^{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}} \]

      2. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      3. lower-pow.f64 N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

      4. exp-diff N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{e^{\log z}}{e^{t}}\right)}}^{y} \]

      5. rem-exp-log N/A

         \[\leadsto x \cdot {\left(\frac{\color{blue}{z}}{e^{t}}\right)}^{y} \]

      6. lower-/.f64 N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{z}{e^{t}}\right)}}^{y} \]

      7. lower-exp.f64 72.0

         \[\leadsto x \cdot {\left(\frac{z}{\color{blue}{e^{t}}}\right)}^{y} \]

   5. Applied rewrites 72.0%

      \[\leadsto x \cdot \color{blue}{{\left(\frac{z}{e^{t}}\right)}^{y}} \]

   6. Taylor expanded in t around 0

      \[\leadsto x \cdot {z}^{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.3%

      \[\leadsto x \cdot {z}^{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.28 \cdot 10^{+141}:\\ \;\;\;\;\left(\left(0.5 \cdot t\right) \cdot \left(\left(y \cdot y\right) \cdot x\right)\right) \cdot t\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{+166}:\\ \;\;\;\;{z}^{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.5 \cdot t\right) \cdot \left(\left(y \cdot y\right) \cdot x\right)\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 19.3% accurate, 54.7× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return 1.0 * x;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return 1.0 * x;
}

Python

def code(x, y, z, t, a, b):
	return 1.0 * x

Julia

function code(x, y, z, t, a, b)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 92.7%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto x \cdot \color{blue}{e^{y \cdot \left(\log z - t\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}} \]

   2. exp-prod N/A

      \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

   3. lower-pow.f64 N/A

      \[\leadsto x \cdot \color{blue}{{\left(e^{\log z - t}\right)}^{y}} \]

   4. exp-diff N/A

      \[\leadsto x \cdot {\color{blue}{\left(\frac{e^{\log z}}{e^{t}}\right)}}^{y} \]

   5. rem-exp-log N/A

      \[\leadsto x \cdot {\left(\frac{\color{blue}{z}}{e^{t}}\right)}^{y} \]

   6. lower-/.f64 N/A

      \[\leadsto x \cdot {\color{blue}{\left(\frac{z}{e^{t}}\right)}}^{y} \]

   7. lower-exp.f64 64.4

      \[\leadsto x \cdot {\left(\frac{z}{\color{blue}{e^{t}}}\right)}^{y} \]

5. Applied rewrites 64.4%

   \[\leadsto x \cdot \color{blue}{{\left(\frac{z}{e^{t}}\right)}^{y}} \]

6. Taylor expanded in y around 0

   \[\leadsto x \cdot 1 \]
7. Step-by-step derivation

8. Applied rewrites 16.7%

   \[\leadsto x \cdot 1 \]

9. Final simplification 16.7%

   \[\leadsto 1 \cdot x \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))